Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 6 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 8 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

January 25, 2022 at 12:01am

#1025331

Changes

You'll be pleased to know that there's only a week left of me doing prompts from "JAFBG"

[XGC].

The world changes pretty quickly. What are some things that have changed in society since you were a kid and what things have stayed the same?

I guess the biggest change is the extinction of the woolly mammoth. Those things used to shake the ground in their herds, but then one day, boom, gone.

Or maybe it was the invention of the wheel. Well, actually, wheels were around before I was born; it's the axle that really changed society by letting us load captured enemies on carts instead of dragging them behind horses.

Okay, okay, fine, something serious.

Society ebbs and flows, and most changes are fads and fashions, superficial stuff. But technology has the potential to bring lasting changes to society, so I'm mostly focusing on technology here. And again, this is a U.S. perspective because that's where I live.

My birth year was a little more than 60 years after the first powered controlled flight. The first 747 took off a few years after I was born. In just a few years, more time will have passed between "now" and my birth than between my birth and the first flight. A lot changed in those first 60-some years, but commercial air travel since then? Apart from a few details, such as trip volume, engine efficiency, security theater, and the willingness to pack us into those damn aluminum tubes like pickles in a jar, it hasn't materially changed since I was born. Oh, we toyed with supersonic jets for a bit, but they never made a lot of commercial sense.

Nor has automobile travel changed much, except for increased safety measures -- interstates were mostly built before I was born.

What I'm getting at is we're using the same modes of transportation as we did in the 60s. The only exception is urban scooter rentals and the like, which hardly count. Other countries introduced higher-speed rail, but that's still just rail. Electric cars are still just cars.

The reason I mention this is that there is one thing that has changed radically, which is communication. The internet has connected all of us in a way transportation doesn't -- for better and for worse.

And even with all the downsides of it, I'd rather have the internet than not. Being able to communicate in real time with people from all over the world? That's very cool. Streaming video on demand? Absolutely cool. Knowledge at my fingertips? Love it (even if I do have to double- and triple- check most of it). Best of all, I can usually avoid the parts of it that suck, such as vertical video, social media and intrusive ads.

I can't let this go without nodding in the direction of the most important U.S. societal change in my lifetime.

On October 14, 1978, President Carter signed federal transportation bill H.R 1337 into law. It included Amendment Number 3534, proposed by Senator Alan Cranston of California, authorizing the home production of wine and beer.

This federal legalization of home brewing created many hobby brewers, some of whom went on to open craft breweries, thus changing society for the better. We don't live in a perfect world, of course, but having microbreweries around at least keeps this timeline from verging into absolute dystopia.

So which was more important, craft breweries, or the internet? I really can't say. I do know that I'd never find as many beers to try if it weren't for the internet... but on the flip side, I'd have less use for the internet if I couldn't use it to find breweries. So... both?

Sure. Why not both.

4 Like • Comment • View 1 comment

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